an algebraic introduction to kapustin-witten theory · an algebraic introduction to kapustin-witten...

An Algebraic Introduction to Kapustin-Witten Theory

Chris Elliott

October 24th to 26th, 2017

Abstract

Kapustin and Witten constructed a family of ”twisted N = 4 gauge theories” in four dimensions in order tobuild a bridge between gauge theory and the geometric Langlands correspondence. In these lectures I’ll introduceN = 4 theories and explain how to derive their twists in a purely algebraic way. I’ll discuss some aspects ofthe quantization of these theories, and explain an application to the theory of singular supports in geometricLanglands. This is joint work with Philsang Yoo.

1 Lecture 1 – Kapustin-Witten Twists: a Global, Classical Description

1.1 Introduction

My goal with this series of lectures is to explain how the family of topological field theories constructed by Kapustinand Witten [KW07] can be mathematically understood in the setting of derived algebraic geometry. This is aworthwhile exercise for at least two reasons. On the one hand, Kapustin and Witten’s work was motivated bythe goal of drawing a bridge between supersymmetric gauge theory and the geometric Langlands conjecture – bycarefully mathematically modelling these theories we can make this connection more precise and fill in gaps in thephysical story, for instance by explaining the appearance of algebraic structures on moduli spaces, and singularsupport conditions in the modern formalulation of geometric Langlands. Similarly the physical approach suggestsnew structures in the geometric Langlands story itself which we’ll discuss in the third lecture. On the other hand,Kapustin-Witten theory provides an interesting but computationally tractable example of the general formalism ofclassical and quantum field theory – for instance it’s close to (but not fully) topological – which we can use as atest example for the more general problem of puzzling out the non-perturbative structure of similar twisted gaugetheories in other dimensions. The work I’ll talk about today is joint with Philsang Yoo [EY15,EY17]. Some partsof these notes are excerpted from those two papers.

I’ll start with a brief summary of the structure of the three talks.

1. In the first lecture I’ll talk about what it means to have a topologically twisted field theory from the point ofview of derived algebraic geometry. This is, I claim, a natural setting in which to talk about classical fieldtheory “non-perturbatively”, that is, including the full data of the fields and action functional of a classicalfield theory. In order to get there, I’ll start with an introduction to the facts about derived algebraic andshifted symplectic geometry that we’ll refer to in the three talks, and then sketch a definition of what I meanby a classical field theory.

Now, from this point of view to twist a classical field theory mean essentially to take the derived invariantswith respect to an odd symmetry. I’ll talk about a nice family of field theories – a certain kind of gaugetheory which makes sense algebraically – in which we can perform this procedure, and tell you how to fit thefamous physical N = 4 4d gauge theory into this family. We’ll then describe the Kapustin-Witten twists ofthis theory, and see some interesting representation theoretic moduli spaces appearing.

1

2 Section 1 Lecture 1 – Kapustin-Witten Twists: a Global, Classical Description

2. In the second lecture I’ll talk about the local structure of these twisted field theories. I’ll compute thelocal observables in these theories and see what happens when we try to quantize them – in fact these localobservables do not admit any quantum corrections. We get something more interesting when we look at whatthese field theories assign to an algebraic curve. I’ll explain an ansatz out of which we obtain the geometricLanglands categories by geometric quantization. I’ll also give a short introduction to what exactly geometricLanglands theory is.

3. In the third and final lecture I’ll explain the results of my paper with Philsang Yoo [EY17]. By investigatingthe action of local observables on the categories of boundary conditions on a curve, and imposing a conditionon the support with respect to this action, we’ll see that the singular support conditions of Arinkin andGaitsgory [AG12] naturally arise. We can motivate these conditions physically, and generalize them to seesome potentially interesting new structures on the geometric Langlands correspondence.

1.2 Some Facts About Derived Symplectic Geometry

In these talks I’m going to be talking about some examples of derived stacks with shifted symplectic structures –these things naturally appear in classical field theory. As such I’ll start with a quick introduction to what thesewords mean. The field of derived algebraic geometry includes a lot of sophisticated theory (most of which I knowvery little about), but today I’m only going to state a few definitions along with one key theorem about shiftedsymplectic structures on mapping spaces. The theory of derived symplectic geometry was developed by Pantev,Toen, Vaqiue and Vezzosi [PTVV13] and we refer to their work for details. Whenever I say “category” I’ll reallymean “∞-category”, but luckily for what I have to say today we won’t need to engage with any of those issues.

So what is a derived stack? I’ll give a sketch definition so that you have something to hold on to, but much moreimportantly I’ll give a list of examples.

Definition 1.1. A prestack is a functor X : cdga≤0 → sSet from commutative dgas over C concentrated in degrees≤ 0 to simplicial sets. A derived stack is a prestack that satisfies a descent condition for the etale topology.

Remark 1.2. For us all derived stacks will be Artin stacks locally of finite presentation. This is a technicalcondition that, for instance, ensures the tangent complex is perfect, hence dualizable, which will be important whenwe talk about symplectic structures.

1. Every classical scheme or stack is an example of a derived stack.

2. Every simplicial set M is an example of a derived stack by taking the constant functor M(R) = M .

3. If S is a commutative dga concentrated in degrees ≤ 0 then there is a derived stack called the spectrum of Swhose R points are given by

Spec(S)(R)k = Homcdga(S,R⊗ Ω•alg(∆k)).

There’s an alternative model for these derived schemes as ringed spaces (X,OX), where (X,H0(OX)) is aclassical scheme and Hi(OX) is a quasi-coherent sheaf for each i > 0.

4. If X and Y are derived stacks then there is a mapping stack Map(X,Y ) whose R points are given by

Map(X,Y )(R)k = HomdStacks(X × Spec(R⊗ Ω•alg(∆k)), Y ).

5. If X is a derived stack and k is an integer then there’s a derived stack called T [k]X – this k-shifted tangentspace of X. We can model it as the mapping stack T [k]X = Map(SpecC[ε], X) where ε is a degree −kparameter satisfying ε2 = 0.

6. If X is a derived stack then one can define the de Rham stack of X by

XdR(R) = X(Rred),

where Rred is the quotient of R by its nilradical. Roughly speaking one should think of XdR as what you getby identifying “infinitesimally close points” of X.


All of the derived stacks I’ll talk about this week will be built using these examples and constructions.

Definition 1.3. We can talk about quasi-coherent sheaves on a derived stack X by taking the limit over all affinederived schemes mapping in, where we set QC(SpecR) = R-mod. An important examples is gives by the cotangentcomplex LX of a derived stack X. For the examples we’ll be working with, the cotangent complex is alwaysdualizable; we call its dual the tangent complex TX .

Once we have the notion of the cotangent complex, it makes sense to talk about p-forms on a derived stack. However,since the tangent complex is a sheaf of cochain complexes, likewise there will be a complex of p-forms.

Definition 1.4. A p-form of degree k on X is a section of degree k (meaning a map of sheaves from C[−k]) of thesheaf of p-forms on X:

ΩpX = Symp(LX [1])[−p].

The de Rham differential is a sheaf map ddR : Ω•X → Ω•+1X [1] which extends the derivative OX → LX as a derivation.

The complex Ω•X is therefore bigraded : there’s an internal differential d of degree 1 coming from the differential onLX , and there’s also the de Rham differential which has degree −1 (even though it increases the weight – the pabove – by 1). A closed p-form of degree k is a section of degree k of the complex

Ω•cl,X = (Ω•X ⊗C C[[u]],d + uddR)

where u is a parameter of degree 2. Note that in contrast to the classical story, being closed is not a condition.Indeed if you compute the kth cohomology of Ω•cl,X you find not a de Rham-closed element, but an element whichis de Rham closed up to the addition of a d-exact form which in turn satisfies a tower of higher coherences. Inparticular there’s a map Ω•cl,X → Ω•X but it’s not injective in any sense.

With this in mind, the definition of a shifted symplectic structure is roughly what you might guess.

Definition 1.5. A k-shifted symplectic structure on a derived stack X is a closed 2-form of degree k which isnondegenerate in the following sense. Any 2-form of degree k defines a morphism TX → LX [k] and any closed2-form defines an associated 2-form: the closed 2-form is non-degenerate if this induced morphism is a quasi-isomorphism.

There are a few ways of building shifted symplectic structures (see [PTVV13] for details and proofs).

1. The k-shifted cotangent space T ∗[k]X of a derived stack always carries a canonical k-shifted symplecticstructure.

2. If G is a complex reductive group, the classifying stack BG carries a canonical 2-shifted symplectic structure.The tangent complex of BG is equivalent to g placed in degree −1 and equipped with the adjoint action ofG. One can verify that the closed 2-forms of degree 2 are exactly the G-invariant pairings g ⊗ g → C, andnon-degeneracy is equivalent to non-degeneracy of the pairing. In other words there’s a canonical 2-shiftedsymplectic structure given by the Killing form.

3. This won’t be too important for this week’s lectures, but it’s worth mentioning that there’s a notion of ak-shifted Lagrangian structure on a morphism of derived stacks f : L → X when X is k-shifted symplectic(there’s no restriction, by the way, that f is an embedding).

Theorem 1.6. If L1 → X and L2 → X are k-shifted Lagrangian then the derived fiber product L1 ×X L2

carries a canonical (k − 1)-shifted symplectic structure.

4. We’ll use the following fact a lot more.

Theorem 1.7 (AKSZ-PTVV). Let X be a k-shifted symplectic derived stack, and let M be a derived stackwhich is compactly oriented of degree n. Roughly speaking this is a perfect pairing on its cohomology, so forinstance closed oriented n-manifolds are examples (by Poincare duality), or n-dimensional smooth Calabi-Yauvarieties (by Serre duality). Then there is a canonical (k − n)-shifted symplectic structure on the mappingspace Map(M,X).


I can give you an idea of roughly what this symplectic structure is: it’s defined via transgression. There’s anatural evaluation map ev : Map(M,X)×M → X, so by pulling back the symplectic form on X we obtain aclosed 2-form of degree k ev∗(ω) on Map(M,X)×M . The degree n compact orientation defines a pushforwardmap from degree k closed 2-forms on the product to degree k − n closed 2-forms on Map(M,X). One canthen demonstrate that this closed 2-form is non-degenerate.

1.3 N = 4 Gauge Theory

1.3.1 Classical Field Theories

So, what is a classical field theory? If you attended my talk yesterday you heard one definition in terms of asheaf of dg Lie algebras with an invariant pairing. The definition I’ll give today will be a global version of thatlocal definition. In physics, a Lagrangian field theory on a space M is modelled by a sheaf of spaces Φ, called thefields, along with a map S to the constant sheaf called the action functional. The classical states of this system aremodelled by the critical locus of S. Passing to the critical locus loses some information about the pair (Φ, S), solet’s instead use a derived version of this construction.

Definition 1.8. Given a derived stack Φ and a function S : Φ → C, the derived critical locus of S is the derivedintersection

dCrit(S) = Φ ∩T∗Φ ΓdS ,

where Φ→ T ∗Φ is the zero section, and ΓdS is the graph of the 1-form dS. Note that T ∗Φ is 0-shifted symplecticand both the 0-section and the graph of a 1-form are Lagrangian, so by Theorem 1.6 the derived critical locus isalways (−1)-shifted symplectic.

Remark 1.9. You might object that if we really want to consider the space of fields on a non-compact spacetimethen the action functional is not typically well-defined (there’s a well-defined Lagrangian density but it doesn’thave a finite integral). This is not a problem for this formalism: even though S isn’t necessarily well-defined itsderivative dS still will be.

We can model a classical field theory by its full derived critical locus with its (−1)-shifted symplectic structure.

Definition 1.10. A classical field theory on M is a sheaf EOM (for the solutions to the equations of motion) ofderived stacks on M with a (−1)-shifted symplectic structure on the global sections EOM(M).

Remarks 1.11. 1. A more sensitive definition might include a (−1)-shifted Poisson structure on the localsections EOM(U) compatible with the sheaf structure.

2. In general the full sheaf is often difficult to construct even without tracking the Poisson bracket. In whatfollows we’ll jut study a single (−1)-shifted symplectic stack at a time, modelling the global sections.

3. To recover the local definition of a classical field theory that I discussed yesterday, one chooses a point inEOM(M) and takes the (−1)-shifted tangent complex. This is automatically a dg Lie algebra, and the shiftedsymplectic structure induces a pairing onthe shifted tangent complex of degree −3. This is what we call theBV-BRST complex of the classical field theory equipped with its antibracket.

1.3.2 N = 4 Super Yang-Mills

The examples we’ll discuss today will be classical field theories defined as twists of a theory called N = 4 superYang-Mills theory in 4-dimensions. Let me try to explain what this theory is and why it might have anything to dowith algebraic geometry. There are two different ways of building this 4d gauge theory. The original constructionuses N = 1 super Yang-Mills theory on R10. I won’t talk about this approach today, although it’s a fun calculation.Instead I’ll describe another construction due to Penrose and Ward via super twistor space (for details we refer tothe book [WW91] of Ward and Wells).


Definition 1.12. N = 4 super twistor space is a super complex algebraic variety whose bosonic part has dimension3. It’s actually quite simple, it’s nothing but

PTN=4 = P(C4|4),

the space of (ordinary, bosonic) lines in a super vector space. Even more concretely it’s the total space of the vectorbundle Π(O(1)⊕4) on CP3. Super twistor space admits a map (sometimes called the Penrose map) down to the4-sphere coming from the map

CP3 → HP1 ∼= S4

whose fibers are the total space of an odd vector bundle over CP1 – the so-called twistor lines (exercise: check this).In particular there’s a map

p : PTN=4 \ CP1 → R4

obtained by throwing out the fiber over ∞. Note: even though the left-hand side is a complex manifold, this mapis not holomorphic for any complex structure on R4.

The Berezinian (the super version of the canonical bundle) of PTN=4 is trivializable. Indeed, one can compute the

Berezinian of any superprojective space CPm|n to be

Ber(CPm|n) ∼= KCPn ⊗ ∧n(O(1)⊕n)∼= O(−m− 1)⊗O(n)∼= O(n−m− 1)

so in this case m = 3, n = 4 so the Berezinian is trivial.

N = 4 super Yang-Mills will arise as the pushforward along p of a classical field theory on super twistor space.Let’s explain what that theory is.

Definition 1.13. One can define algebraic Chern-Simons theory on any super Calabi-Yau smooth super 3-fold,i.e. a smooth super variety Y whose bosonic part has dimension 3 with trivialized Berezinian. As a (−1)-shiftedsymplectic derived stack it’s defined as

BunG(Y ) = Map(Y,BG).

This is (−1)-shifted symplectic whenever Y is compact by Theorem 1.7 since BG is 2-shifted symplectic and Y is3-oriented.

Remark 1.14. This is the origin of the algebraic structures that we’ll use in Kapustin-Witten twisted N = 4theory: one usually says that one obtains 4d N = 4 theory as the pushforward of holomorphic Chern-Simonstheory on super twistor space, i.e. considering the complex analytic stack of holomorphic G-bundles. We observethat this stack admits a natural algebraic structure which carries over to provide an algebraic structure to theKapustin-Witten topological twists.

Remark 1.15. I’m lying a little bit: the theory obtains by pushing forward holomorphic Chern-Simons theoryto R4 isn’t exactly the same as N = 4 super Yang-Mills, it’s only the anti-self-dual sector of that theory. Boels,Mason and Skinner [BMS07] demonstrated that the holomorphic Chern-Simons theory can be modified in orderto produce the full N = 4 theory under compactification, but we won’t need to worry about this distinction – thedifference between the full N = 4 theory and the anti-self-dual N = 4 theory vanishes after performing the twist.

Remark 1.16. We run into trouble when we try to define untwisted N = 4 super Yang–Mills theory non-perturbatively via compactification along the twistor fibers, because the Penrose map p is not holomorphic forany complex structure on R4. As such, a Zariski open set U ⊆ C2 does not lift to a Zariski set p−1(U) ⊆ PT \ CP1.This is not a problem in the analytic setting; any open set in a complex manifold admits a canonical complex struc-ture, but generally not an algebraic structure. It is not particularly surprising that we encounter such problems:there’s no reason that a metric-dependent theory like untwisted N = 4 gauge theory should admit a descriptionpurely in terms of algebraic geometry. What we’ll actually do is use this algebraic theory to define twisted versionsof N = 4 which will make sense as theories on R4.


Remark 1.17. Another related issue is the fact that when we remove the twistor line over the point at ∞ thesuper twistor space is no longer compact, so we no longer have an AKSZ shifted symplectic structure (at most we’llhave a shifted Poisson structure). Similarly, we won’t really use the shifted symplectic structure for the untwistedN = 4 theory, only the structure we obtain after turning on a twist, at which point we can define the theory onmore general compact complex surfaces and avoid this issue of non-degeneracy.

1.3.3 The N = 4 Supersymmetry Algebra

The reason this theory is special is that it carried an action of the N = 4 supersymmetry algebra. I talked aboutsupersymmetry algebras in general yesterday, but I’ll say now what this specific example looks like. For simplicityI’ll just describe the complexified supersymmetry algebra – the data of its real form won’t be so important.

Definition 1.18. There is an exceptional isomorphism so(4;C) ∼= sl(2;C) ⊕ sl(2;C). The 4-dimensional complexspin representation of so(4) is the sum S+ ⊕ S− of the fundamental representations of the two copies of sl(2;C).The 4-dimensional fundamental representation of so(4;C) is identified with the tensor product S+ ⊗ S−.

The complexified N = 4 supersymmetry algebra is the complex super Lie algebra

AN=4 = ((so(4;C) nC4))× sl(4;C)) n Π(S+ ⊗W ⊕ S− ⊗W ∗)

where W is a complex vector space of dimension 4. The algebra sl(4;C) is the algebra of R-symmetries: it acts onW as the fundamental representation. There is a bracket between the two summands of the fermionic part valuedin C4 by the evaluation pairing W ⊗W ∗ → C and the canonical identification S+ ⊗ S− ∼= C4.

Remark 1.19. On the level of the supersymmetry algebra there is a full GL(4;C) of R-symmetries acting on W ,but only the SL(4;C) subgroup acts on the N = 4 theory. There’s a nice way of seeing this from the super twistorspace point of view: only SL(4) preserves the trivialization of the Berezinian which we had to fix in order to definethe shifted symplectic structure on the algebraic Chern-Simons theory moduli space.

In order to define a twist of a theory with an action of this supersymmetry algebra, we identify a supercharge Q(i.e. a fermionic element of the algebra) such that Q2 = 0. Equivalently Q should span a one-dimensional fermionicsubalgebra of AN=4. Let’s discuss the supercharges we’ll use.

Definition 1.20. A holomorphic supercharge is a square-zero Q such that the image of [Q,−] is half-dimensionalinside of C4. We can easily find a holomorphic supercharge by taking a rank one element of the summand S+⊗W ,i.e. an element of the form Qhol = α ⊗ w where α ∈ S+ and w ∈ W . All such elements square to zero and areholomorphic.

Definition 1.21. A topological supercharge is a square-zero Q such that the image of [Q,−] is all of C4.

Let’s analyse the topological supercharges that arise as “further twists” of a holomorphic supercharge Qhol. Thatis topological supercharges of the form Qhol +Q′ where Q′ commutes with Qhol but is not the image of Qhol underome bosonic symmetry. Equivalently these are topological supercharges coming from the [Qhol,−]-cohomology ofAN=4.

It turns out there is, up to R-symmetry, a CP1-family of such topological further twists. To write it explicitly,choose bases 〈α1, α2〉, 〈β1, β2〉 and 〈, w1, w2, w3, w4〉 for S+, S− and W respectively. Let Qhol = α1 ⊗w1. Then thefamily of further topological twists is given by

Qhol + λα2 ⊗ w2 + µ(β1 ⊗ w∗3 − β2 ⊗ w∗4)

where (λ : µ) is a point in CP1.

Remark 1.22. We should say something briefly about the Kapustin-Witten twisting homomorphism: the blockdiagonal embedding φKW : sl(2;C)⊕ sl(2;C)→ sl(4;C). We can use this to modify the SO(4) action on the N = 4theory in order to define the twisted theories on more general curved manifolds (the untwisted theory is only


defined along with its supersymmetry action on framed manifolds). We should note however that the holomorphicsupercharge, and in fact all but the point QB = α1 ⊗ w1 − α2 ⊗ w2, are now invariant for the full twisted SO(4)-action, but only for U(2), or for SO(2)× SO(2). So they can be defined on arbitrary complex manifolds but not infull generality.

1.4 Global Twists of Classical Field Theories

Before we talk about our key examples, let’s talk about what it means to twist a derived stack (building on thebetter understood notion of a perturbative twist, which I talked about yesterday but which I’ll quickly reviewtoday). Suppose we’re given an action of the group C× n C[−1], i.e. an odd symmetry Q such that Q2 = 0, andan action α of C× such that Q has weight 1. In supersymmetric field theories as we’ve discussed in the example of4d N = 4, one has an action of the group of supersymmetries, and one can often find many copies of C× n C[−1]embedded inside it, as we just discussed. If you take the tangent complex TpM at any point in the moduli spaceof classical solutions, the action of this supergroup is the same as a dg-structure, i.e. a grading (the α-weight) anda differential (Q) of degree 1.

Definition 1.23. To twist this tangent complex means to take the total complex for its internal grading anddifferential and this new grading and differential.

Remark 1.24. We can talk about these perturbative twists from another point of view. One can think of asupersymmetric theory with a chosen square-zero supercharge Q as a family of theories living over the classifyingspace B(ΠC), or equivalently a theory with an action of the ring C[[t]] where t is a fermionic parameter of degree1. The C× action allows us to define a twisted theory by inverting the parameter t – or equivalently restrictingto the formal punctured disk – then taking C×-invariants. This is equivalent to the recipe I just described (seeCostello [Cos13] where this construction was first given in this kind of language).

One can define twists of the whole derived stack M in the following setting. Fix a base derived stack B, andnilisomorphisms π : M B: σ (that is, maps that induce isomorphisms after taking H0), where σ is a section of π.That is, M is a pointed formal moduli problem over B. In our examples coming from gauge theory, B = BunG(X)– we call such theories formal algebraic gauge theories. One can then show the following.

Proposition 1.25. If π : M B: σ is a pointed formal moduli problem over B, and M admits an action (α,Q)of C× n C[−1] so that σ is equivariant for the trivial action on B, then there exists a formal moduli problemMQ ← B : σQ under B whose relative tangent complex at each point b ∈ B is obtained by twisting the relativetangent complex of σ.

Remark 1.26. How does this work? Well, Gaitsgory and Rozenblyum [GR] prove that pointed formal moduliproblems are uniquely determined by their relative tangent complex, as a sheaf of (dg) Lie algebras, and unpointedformal moduli problems (where there is only a map σ, not π) are determined by the tangent Lie algebroid. One canuse the perturbative definition above to twist the relative tangent complex, possibly breaking the map π, to obtaina new Lie algebroid, then use Gaitsgory and Rozenblyum’s equivalence to obtain a new formal moduli problem.

Example 1.27. If S is a smooth scheme, the derived stack T [1]S has a canonical family of twists, where Q acts asthe differential λ × id : TsS[1] → TsS on the tangent complex. The twist with respect to this Q is denoted Sλ-dR:when λ = 1 this is the de Rham stack of S.

Our starting point is the following holomorphically twisted N = 4 theory. This is the minimal twist of N = 4 superYang-Mills theory that can be defined algebraically.

Theorem 1.28. The holomorphically twisted N = 4 Yang-Mills theory on a smooth proper complex surfaceX is the derived stack T ∗[−1]HiggsG(X), with its canonical shifted symplectic structure. Here, HiggsG(X) =Map(T [1]X,BG) is the stack of Higgs bundles on X.

Proof Outline. Let’s just give a rough idea of how this is proven. We compute the twist at the level of twistor space– the twisted theory turns out to be localized on a section of the twistor fibration which makes it easy to compute

8 Section 2 Lecture 2 – A Local Description in 2 and 4d, and Quantization

the pushforward to C2, and to define the theory on more general complex surfaces. All we need to do is to viewholomorphic Chern-Simons on twistor space as a sheaf of super dg Lie algebras over BunG(CP3), and to computethe cohomology of this sheaf with respect to Qhol and an appropriate R-symmetry circle. The resulting cohomologylooks like the complex Ω•,•(C2, g) – which is the shifted tangent complex to the moduli space of Higgs bundles –plus a shift of its dual.

Now, observe that this can be written in a more symmetrical way! Since X is smooth and proper it has anorientation of dimension 4, and BG has a 2-symplectic structure given by the Killing form, which means, bythe AKSZ construction of Pantev-Toen-Vaquie-Vezzosi [PTVV13] there’s a (−2)-shifted symplectic structure onHiggsG(X). We can therefore write

T ∗[−1]HiggsG(X) ∼= T [1]Map(T [1]X,BG).

Now, this has a P1-family of deformations, by deforming either appearance of the Dolbeault stack T [1]. This leadsto the main theorem of [EY15].

Theorem 1.29 (E-Yoo). The P1 of twists of the holomorphically twisted N = 4 theory coincide with the P1 oftopological twists constructed by Kapustin and Witten.

What does this mean? Well, one can twist a supersymmetric field theory using any square 0 supercharge. Weexplicitly computed the action of the supersymmetry algebra on the N = 4 theory, and hence on its holomorphictwist, and proved that the P1 of deformations described above coincides with the action of Kapustin and Witten’sfamily of topological supercharges.

Examples 1.30. The most important two points in this P1 family are (1 : 0) and (0 : 1), which we call the A- andB-twists respectively. The spaces we obtain in these two twists are HiggsG(X)dR and T ∗[−1]FlatG(X).

2 Lecture 2 – A Local Description in 2 and 4d, and Quantization

Yesterday I explained how to associate derived moduli stacks to the Kapustin-Witten twists of N = 4 gauge theory:they’re the moduli spaces of classical solutions to the equations of motion after turning on the appropriate twist.Today I’ll start to talk about quantization. First I’ll discuss the perturbative quantization (that is, the quantizationof the P0-algebra of local observables). We’ll see that this isn’t very interesting, and say something about whatthe quantum theories should assign to an algebraic surface Σ. Along the way we’ll talk a little bit about geometricLanglands.

2.1 The A- and B-twisted Theories Locally

So we concluded our discussion yesterday by computing the moduli stacks our Kapustin-Witten twisted theoriesassign to a complex surface X. Let’s continue by discussing what these theories look like in positive codimension.

Let’s begin with the B-twist. We write D = SpecC[[t]] and D× = SpecC((t)) for the formal disk and the formalpunctured disk respectively.

Proposition 2.1. The Kapustin-Witten B-twisted assigns the following derived stacks to spaces in positive codi-mension:

EOMB(D× × Σ) ∼= T ∗FlatG(Σ)

EOMB(D× Σ) ∼= T ∗[1]FlatG(Σ)

EOMB(D× D) ∼= T ∗[3]BG ∼= g∗[2]/G.


To see why this is the case, remember that we compute the twist of an algebraic gauge theory by identifying theuntwisted theory with a shef of Lie algebras over BunG(Σ). Running this argument starting with the holomorphicallytwisted N = 4 gauge theory, the sheaf of Lie algebras in question splits as a sum of the Lie algebra determining themoduli of Higgs bundles (namely the complex Ω•,•(X; gP )) plus a copy of its dual, shifted by −1. We can identifythe dual of the complex of (p, q) forms on each of these spaces with a shift of the same complex, but where the shiftdepends on the space we choose. We thus obtain, in the twist, a shifted cotangent space where the shift dependson the codimension in which we’re working.

Remark 2.2. We can identify the spaces that occur in the A-twist or in a mixed twist in a similar way. In theA-twist things are less interesting because the cotangent disappears after taking the de Rham stack, but it’s naturalto think of EOMA(D×Σ) as (HiggsG(Σ))dR, and EOMA(D×D) as (BG)dR. In what follows we’ll focus mainly onthe B-twist.

Now, with this calculation in hand, let’s talk about the classical observables in our twisted gauge theories (this willprovide an example of the formalism I discussed on Monday). The classical observables are nothing but functionson the space of solutions to the equations of motion.

Definition 2.3. The local classical observables in the B-twisted theory are given by the algebra of functions on thelocal solutions to the equations of motion. Namely

ObsclB(B4) = O(g∗[2]/G) ∼= O(h∗[2]/W ).

What sort of structure does this algebra have, and why should we expect this structure from the point of view offield theory? Well, since g∗[2]/G ∼= T ∗[3]BG is 3-shifted symplectic, its algebra of functions has a 3-shifted Poissonalgebra structure. This is natural from the point of view of classical topological field theory, as we’ll explain, butin fact, for degree reasons, this Poisson structure becomes trivial at the level of global functions of the stack.

Definition 2.4. A Pn-algebra (in cochain complexes) is a commutative dga A equipped with a bracket , : A⊗A→A[1−n] called an 1−n-shifted Poisson bracket, which is graded anti-symmetric and satisfies a graded Jacobi identity,and is a biderivation for the product.

Remark 2.5. In particular, the algebra of functions on an n-shifted symplectic stack always comes equipped witha Pn+1-algebra structure.

These Pn-algebra structures show up naturally in topological theory because of the following theorem. Recall thatthe algebra of local observables in a topological field theory can naturally be given the structure of an En-algebra.

Theorem 2.6 (Poisson additivity [Saf16]). The data of a Pn-algebra is equivalent to the data of a P0-algebra inthe category of En algebras. More succintly there is an equivalence of operads Pn ∼= P0 ⊗ En

As such, the local observables in a classical topological field theory should always have a Pn-algebra structurecoming from the (−1)-shifted symplectic structure on the moduli space of solutions to the equations of motion andthe factorization structure.

Remark 2.7. We get something very similar on the A-side: the algebra of local observables looks like

ObsclA(B4) = O((BGdR) ∼= H•dR(BG).

The de Rham cohomology of BG is indeed generated by the Cartan algebra h (identified with the space of invariantpolynomials), but not generally concentrated entirely in degree 2, so there is an ungraded isomorphism between thelocal observables in the A- and B-twisted field theories. The algebra is however concentrated in non-negative evendegrees, which means that there cannot be a non-trivial 3-shifted Poisson bracket.

2.2 Quantization of the Observables

So, we’ve found that in the B-twisted theory the classical local observables are given by the algebra O(h∗[2]/W )equipped with the trivial P4-structure. Let’s now talk about what it means to quantize this algebra. In this local,


topological context, the problem of quantization has been studied in detail, so we’ll be able to say something veryprecise.

We gave a concrete definition of a Pn-algebra above, but if n is at least 2 one can instead characterize the Pn-operad as the cohomology of the En-operad. In particular, given an En-algebra A one can produce an Pn-algebraby taking its cohomology. To put it more precisely we can say the following. For an En-algebra A, by definitionwe have a map Emb(

∐I B

n, Bn) × AI → A. For I = 1, 2, we have a map Sn−1 → Emb(Bn∐Bn, Bn) by

considering the first disk fixed at the origin. This gives a map Sn−1 ×A2 → A. Taking cohomology, we get a mapH•(Sn−1)⊗H•(A)⊗2 → H•(A). Thinking of the nontrivial class in Hn−1(Sn−1), we have a map H•(A)⊗2[n− 1]→H•(A), or (H•(A)[n− 1])⊗2 → H•(A)[n− 1].

Theorem 2.8 (Cohen [Coh76]). Let A be an En-algebra. Then the above map on H•(A) induces a Lie bracket ofdegree 1− n on H•(A). Moreover, if n > 1, then H•(A) is a Pn-algebra.

In general, to quantize a P0-factorization algebra means the following. There is an operad over the formal diskD~ called BD0 (short for Beilinson-Drinfeld), whose fiber over ~ = 0 is equivalent to P0 and whose generic fiber isequivalent to E0. A quantization of a P0-factorization algebra is a lift to a BD0-factorization algebra which recoversthe original theory by evaluating at ~ = 0. In the topological case we can put it a lot more simply. To quantize anPn-algebra Acl will therefore mean to find a lift to an En-algebra Aq such that H•(Aq) ∼= Acl as a Pn-algebra.

Surprisingly, the process of going from an En-algebra to a Pn-algebra by taking cohomology doesn’t lose anyinformation for n ≥ 2. This is articulated by the following formality result of the En operad.

Theorem 2.9. [Toe13, Corollary 5.4] For n ≥ 0, if X = SpecA, then the dg Lie algebra CEn+1(X)[n + 1] isnon-canonically quasi-isomorphic to the dg Lie algebra Pol(X,n)[n + 1] of n-shifted polyvector fields. The quasi-isomorphism depends on the choice of a Drinfeld associator.

Here Pol(X,n) = O(T ∗[n + 1]X) is the Pn+2-algebra of shifted polyvector fields. Its shift Pol(X,n)[n + 1] is thedg Lie algebra controlling deformations of the Pn+1-algebra structure on A. On the other side, CEn+1(X) is theEn+1-Hochschild cochain complex, which is an En+2-algebra. Its shift CEn+1 [n + 1] has the structure of a dg Liealgebra controlling the deformations of the En+1-algebra structure on A. So, this formality theorem is saying thatthe space of En+1-deformations of an algebra is equivalent to the space of Pn+1-deformations.

Now let’s go back to our example of the B-twisted theory. In order to understand quantizations of the (trivial)P4-structure on our local classical observables, or equivalently E4-deformations of this trivial structure, we have tocompute the dg Lie algebra Pol(h∗[2]/W, 3)[4]. More precisely, deformations are given by Maurer-Cartan elementsof this dg Lie algebra. However, Pol(h∗[2]/W, 3)[4] = O(T ∗[4]h∗[2]/W )[4] is concentrated in even degrees, so therecan’t be any Maurer-Cartan elements (which live in degree 1). We conclude the following.

Proposition 2.10. The only quantization of the P4 algebra of classical observables in the B-twisted N = 4 theoryis the trivial quantization. That means we can view O(h∗[2]/W ) as the algebra of quantum observables in theB-twisted theory.

This is a little disappointing: the factorization algebras associated to the Kapustin-Witten theories are very boring.By passing from g∗[2]/G of (BG)dR to its affinization we threw away too much information. In the rest of thislecture we’ll describe a less precise ansatz that describes a non-trivial part of the quantum Kapustin-Witten theories,but we’ll see tomorrow that this story requires a correction in terms of this boring algebra of quantum observables,so this calculation will be important even for interesting quantum aspects of the theory!

2.3 Geometric Quantization

2.3.1 The Geometric Langlands Correspondence

I won’t have time to really do justice to this large subject today. I’ll try to give an impression of what sort ofobjects are studied, and what relationships they’re expected to have, and then we’ll observe a connection with the


moduli spaces we’ve been discussing yesterday and today.

Fix a reductive complex algebraic group G and a smooth proper complex algebraic curve Σ. The geomet-ric Langlands program consists of several related conjectures about categorified harmonic analysis on the stackBunG(Σ) = Map(Σ, BG), the moduli stack of algebraic G-bundles on Σ. At its core, the conjecture says thatobjects in the category D(BunG(Σ)) of D-modules on BunG(Σ) can be decomposed according to a nice basis, intoeigenvectors for certain natural operators, and those eigenvectors are parameterised by a moduli space built out ofthe Langlands dual group G∨. So the geometric Langlands conjecture is a kind of non-abelian, categorical Fouriertransform.

Every complex reductive group G has a Langlands dual group G∨, the unique group whose roots are the corootsof G, whose characters are the cocharacters of G, and vice versa. For instance, GLn and SO(2n) are self-dual, SLnand PGLn are dual to one another, as are SO(2n+ 1) and Sp(n).

Definition 2.11. The dual space to BunG(Σ), whose points parameterize eigensheaves on BunG(Σ), is the derivedstack FlatG∨(Σ) = Map(ΣdR, BG) is the derived moduli stack of flat G∨-bundles on Σ. The word “derived” starts

to matter here, for instance if Σ = P1 and G = Gm, FlatGm(P1) ∼= pt×A1 pt×BGm: if you don’t take the derivedstack you lose that first factor.

The following conjecture isn’t the oldest form of the geometric Langlands conjecture, but it’s the more modern,categorical form that appears in the work of Kapustin and Witten.

Conjecture 2.12 (The “Best Hope”). There is an equivalence of categories

D(BunG(Σ)) ∼= QC(FlatG∨(Σ),

which intertwines natural symmetries on the two sides.

Remarks 2.13. 1. A mnemonic for thinking about the geometric Langlands conjecture, at least for G = GLn,is as follows.

The category of flat bundles on the space of vector bundles is equivalent to the category of vector bundleson the space of flat bundles.

This should look familiar when you think about the CP1 of topological twists of the holomorphic theory:there were two spots where a Dolbeault stack could be deformed to a de Rham stack, so the two dual twistslooks like the Dolbeault stack of the moduli of flat bundles, versus the de Rham stack of the moduli of Higgsbundles.

2. If G is abelian, the conjecture is actually a theorem, independently due to Laumon [Lau96] and Rothstein[Rot96]. They realise the equivalence as a twisted version of the Fourier-Mukai transform.

3. Unfortunately, this “best hope” conjecture is false as long as G is non-abelian, one can see this even ifΣ = CP1 [Laf12]: the category QC(FlatG∨(Σ) on the B-side is too small (in a precise sense – the geometricEisenstein series functors fail to preserve compact objects). Arinkin and Gaitsgory [AG12] proposed a correctedform of the geometric Langlands conjecture by enlarging this category in a minimal way. I’ll come back tothis conjecture in tomorrows lecture.

Conjecture 2.14 (Arinkin-Gaitsgory). There is an equivalence of categories

D(BunG(Σ)) ∼= IndCohN (FlatG∨(Σ),

which intertwines natural symmetries on the two sides.

2.3.2 Categories of Boundary Conditions

Let’s connect that geometric Langlands story to our family of topological twists by proposing the following ansatz.

12 Section 3 Lecture 3 – The Category of Boundary Condition, Vacua and Singular Supports

Claim (Categorical Geometric Quantization). Given a 4d classical topological field theory which is a cotangenttheory – meaning the moduli spaces of solutions to the equations of motion are equivalent to shifted cotangentspaces – a good model for the category of boundary conditions along a surfaces Σ is provided by the category ofsheaves on the stackM(Σ), where EOM(Σ) ∼= T ∗[1]M(Σ) is the shifted cotangent space of solutions to the classicalequations of motion.

Remarks 2.15. 1. This ansatz is motivated by the usual story of geometric quantization, where the Hilbertspace of a quantum field theory along a manifold of codimension 1 is modelled by the space of sections of theprequantum line bundle which are constant along the leaves of a polarization. In the case where the phasespace is actually a cotangent bundle there’s a canonical choice of polarization: the Hilbert space is modelledby functions on the base of the cotangent space.

2. There are plenty of subtleties in ordinary geometric quantization which I’m not addressing here. One questionwhich I haven’t answered is: what exactly do I mean by the “category of sheaves”? This is a choice that wehave to make when describing our model. For reasons we’ll discuss in more details tomorrow, in the B-twistedtheory I’ll consider the category of ind-coherent sheaves, but from the point of view of S-duality it’s not clearwhat the matching choice should be on the A-side. I’ll come back to this question.

I’ll conclude today by applying this ansatz in the case of the Kapustin-Witten twisted N = 4 theories.

• In the case of the B-twist, we have EOMB(Σ) ∼= T ∗[1]FlatG(Σ), which means

ZB(Σ) = IndCoh(FlatG(Σ)).

• For the A-twist, we have EOMA(Σ) ∼= (HiggsG(Σ))dR∼= T ∗[1](FlatG(Σ))dR, so our ansatz gives

ZA(Σ) = IndCoh(BunG(Σ)dR) = D-mod(BunG(Σ)).

So we’re seeing the two categories appearing in the (best hope version of) the geometric Langlands conjecture! Ihaven’t really talked about S-duality in this lecture series, but Kapustin and Witten’s program revolves aroundthe fact that dual twists of N = 4 gauge theory, with dual gauge groups, are interchanged by a certain duality ofquantum field theories. In particular this duality should yield an equivalence of the categories of boundary conditionson a curve Σ, which leads Kapustin and Witten to conclude that the geometric Langlands correspondence arises asa consequence of S-duality.

In tomorrow’s lecture I’ll explain how the Arinkin-Gaitsgory singular support condition fits into this physical story.

3 Lecture 3 – The Category of Boundary Condition, Vacua and Sin-gular Supports

My plan today is to conclude this lecture series with a discussion of the connection between the singular supportconditions of Arinkin and Gaitsgory [AG12] and the Kapustin-Witten twisted gauge theories we’ve been discussing.I’ll argue that the singular support conditions occur by considering support conditions for the action of the localobservables that we discussed yesterday on the categories of boundary conditions. This story comes from my jointwork [EY17] with Philsang Yoo.

With this in mind I’ll start out with some background, on how to think about the action of the local observables onthe categories of boundary conditions, on what exactly these singular support conditions are, and then we’ll discusshow the two notions are related. I’ll conclude today with some conjectures and open questions.


3.1 Local Operators and the Moduli of Vacua

Let’s begin today by talking about the action of an E2-algebra on a category. This will be important because itnaturally shows up in the physical story: there’s an E2-action of the algebra of local observables on the category ofboundary conditions assigned to a surface in any 4d topological field theory.

Choose any dg-category C. There’s automatically an action of the monoidal category HC•(C)-mod on C, and thisaction is universal – if you like you can take this universal property as a definition of the Hochschild cochains. Forany cdga A, an action of the category A-mod on C is equivalent to a homomorphism A→ HC•(C).

Example 3.1. Suppose C is the category of boundary conditions in an n-dimensional topological quantum fieldtheory along a smooth compact manifold of dimension n − 2 (for instance, in the context of extended functorialTQFT). There’s always an action of the algebra A of local observables of the field theory on C. In a topologicalfield theory the local observables A naturally form an En-algebra, but the cohomology H•(A) admits an ordinarygraded commutative product.

We think about this action as follows. Let B be the category of boundary conditions along M in a topologicalfield theory, and let F be an object in B. The algebra EndB(F) describes the space of local observables in thebulk-boundary theory associated to think boundary condition, and there’s a tautological map from bulk observablesinto bulk-boundary observables for any choice of boundary condition.

On the other hand, and in the other extreme, we should view the Hochschild cochains as the 2d local operators:that is, the algebra of operators in the 2d theory obtained from compactification along M . Another way of seeingthe action of local operators is by observing that any local operator can be viewed as a 2d local operator (if youlike, this is part of the factorization structure).

If a commutative algebra A acts on a category C, the support of an object in C is a closed subset of SpecA. Thishas a natural physical meaning in the case where A is the algebra of local observables in a topological field theory.It goes like this.

Definition 3.2. States in a quantum field theory on Rn with algebra Obsq(Bn) of local observables are functionalsφ : Obsq(Bn) → R. A state φ is a vacuum state if it translation invariant and satisfies the cluster decompositionproperty, which says O1 on Br1(0) and O2 on Br2(0), we have

(O1 ∗ τx(O2))(φ)−O1(φ)O2(φ)→ 0 as x→∞

where τx denotes the translation of an observable by x ∈ Rn. In a topological field theory this just says that φ is aring homomorphism, so vacuum states are nothing but (closed) points in the spectrum Spec(Obsq(Bn)).

So, to summarise what we just explained, in a 4d topological field theory, the category Z(Σ) assigned to a surfaceΣ is acted on by the algebra of local observables, so we can talk about the support of an object in the moduli spaceof vacua. Let’s talk about what this support means.

Remark 3.3. We motivate this definition in the following way. Objects in the category Bv of boundary conditionswith support v are objects F of B so that EndB(F) is supported at v, just as in the previous subsection. Whatdoes this mean from the point of view of vacua? Well, we should think of the space EndB(F) as the phase space ofour topological field theory coupled to the boundary condition F . Indeed, in general, the hom space HomB(F1,F2)can be interpreted as the space of states on a strip Y n−2 × [0, 1] with boundary conditions F1 and F2 on the twoboundary components. In particular we obtain EndB(F) by putting the boundary condition F at both boundarycomponents. Alternatively (assuming we’re working in a topological context) we can view this as the space ofobservables on Y times a half disk D = (x, y) : x2 + y2 ≤ 1, x > 0, with boundary condition F on the closed edge(see Figure 1). That these descriptions are equivalent is a version of the state-operator correspondence.

From this point of view there’s clearly a map from the algebra A of observables in the bulk to the algebra EndB(F)of observables in this coupled bulk-boundary system by the inclusion of those observables supported at a smallball in the interior of D × Y . In particular using this inclusion one can evaluate elements of EndB(F) at vacua


in V = SpecA. Thus, from this point of view, an object F survives the restriction if the bulk-boundary phasespace EndB(F) is supported at v, in other words if the bulk-boundary system is acted on non-trivially by thoseobservables that only depend on a small neighborhood of v ∈ V. Then objects of the restricted category Bv arethose boundary conditions that can “see” the vacuum state v.

F

×Y

A → EndB(F)

Figure 1: The action of A on F is mediated by a whistle cobordism with the given boundary condition.

3.1.1 Vacua in the B-Twisted Theory

Having introduced the notion of vacua in general, let’s talk about our specific example: the Kapustin-Witten B-twist.Recall that the algebra of classical observables is equivalent to O(h∗[2]/W ) with a trivial shifted Poisson bracket.We argued yesterday that this algebra doesn’t receive any quantum corrections, so it’s exact at the quantum level.The moduli space of vacua is therefore just the spectrum of this algebra:

Vac = h∗[2]/W,

or to put it another way, the affinization of the moduli space of local solutions to the equations of motion. Becauseclassically this derived stack only has one point, there’s only one support condition we can consider: we can considerthe full subcategory of boundary conditions supported at the point 0.

As we’ll see shortly, this condition is actually very interesting, but we can go a little further: it’s possible to regradethe moduli space of vacua in order to remove this shift by 2, after which we can talk about objects supported atany point in h∗/W . There are two ways of doing this, an easy way and a hard way. The easy way is to work in thesetting of Z/2-graded derived algebraic geometry (instead of Z-graded), in which context the even shift obviouslyvanishes. Let’s say something about the more subtle regrading procedure, although it’s a bit of a technical aside soI won’t spend too much time on it.

Recall that when we explained what it meant to twist a derived stack – in particular in Remark 1.24 – we couldview the untwisted theory with its Q-action as a family of theories over B(ΠC). We could then define the twist byinverting th parameter t and taking C×-invariants. What is this is too restrictive: let’s not take those invariants,but instead keep the parameter t in place, though we might restrict to the bosonic part generated by t2 in order toreplace a fermionic degree 1 parameter with a bosonic degree 2 parameter.

If we consider this new theory, the new algebra of local observables looks like O(h∗[2]/W )((t)). It doesn’t makesense to take Spec of this ring because it’s not concentrated in degrees ≤ 0, but we can still talk about its action


on the category of boundary conditions, which now has form IndCoh(FlatG(Σ))⊗Vect C((t))-mod, and the supportof a boundary condition in Spec of H0 of the ring of local operators. This is where the regraded scheme appears:

SpecH0(O(h∗[2]/W )((t))) ∼= h∗/W

now with no shift.

3.2 Singular Support Conditions

3.2.1 What Singular Support Means

Now, let’s move on to the other ingredient of today’s talk: singular support conditions as defined by Arinkin andGaitsgory. Let’s focus on the example where the category C is the category IndCoh(X) of ind-coherent sheaves onsome space X (more precisely, a derived stack). Singular supports are defined using the action of a certain algebraO(Sing(X)) that maps into the Hochschild cochains.

Definition 3.4. Let X be a derived stack. The scheme of singularities of X is the classical part of the −1-shiftedcotangent space

Sing(X) = (T ∗[−1]X)cl.

Suppose X is a (quasi-smooth) affine derived scheme. Then there’s a canonical algebra map (that doesn’t respectthe grading – there’s a degree shift that we’ll mention again later) O(Sing(X))→ HC•(X).

Definition 3.5. If Y is a closed conical subset of Sing(X) then the category of sheaves on X with singular supportin Y is the tensor product

IndCohY (X) = IndCoh(X)⊗QC(Sing(X)) QC(Sing(X))Y .

If X is a more general derived stack then we can define the category of sheaves with singular support in Y ⊆ Sing(X)via smooth descent (take a limit over all smooth maps Z → X whose source is an affine derived scheme).

Example 3.6. From the point of view of quantum field theory, the category IndCoh(X) describes a completedversion of the category of boundary conditions in the 2d B-model with target X. Singular support conditions willadmit a nice physical description from this point of view which we’ll explain shortly – one should interpret thealgebra O(Sing(X)) as the algebra of local operators in the 2d B-model (up to one of these ubiquitous degree shiftsby two).

So I’ve nearly explained to you what the category IndCohNG∨ (FlatG∨(Σ)) is. All I have to do is tell you what thesubsetNG∨ ⊆ Sing(FlatG∨(Σ)) is. First I’ll describe the space of singularities of FlatG∨(Σ). This is a straightforwardcomputation using the fact that the tangent complex to FlatG∨(Σ) is the de Rham complex of Σ with coefficientsin (g∨)∗, shifted down in cohomological degree by one.

Proposition 3.7. The space Sing(FlatG∨(Σ)) is the (classical) moduli stack whose closed points are triples (P,∇, φ)where (P,∇) is a classical point of FlatG∨(Σ) (i.e. a G∨-bundle with flat connection) modulo gauge transformations,and φ is a flat section

φ ∈ H0∇(Σ; (g∨)∗P )

of the coadjoint bundle of P . We call this space ArthG∨(Σ) – the stack of G∨-Arthur parameters of Σ.

Definition 3.8. The global nilpotent cone NG∨ ⊆ ArthG∨(Σ) is the substack consisting of Arthur parameters(P,∇, φ) where the value φx of φ at a point x ∈ Σ is nilpotent as an element of the dual Lie algebra (g∨)∗. Thiscondition doesn’t depend on the choice of point x because the section φ was a flat section.

So now we know what the Arinkin-Gaitsgory category IndCohNG∨ (FlatG∨(Σ)) means. It’s the category of ind-coherent sheaves whose singular support lies in the global nilpotent cone. My goal for the remainder of this talkwill be to explain how this condition arises from quantum field theory.


3.2.2 Singular Support as a Vacuum Condition

Naıvely, we describe the category of boundary conditions in the B-twisted theory by Coh(FlatG(Σ)), or rather byits completion IndCoh(FlatG(Σ)), as in the usual description of the B-model. The action of local observables onthis category has a nice description in terms of geometric representation theory. If we choose a point x ∈ Σ thecategory IndCoh(FlatG(Σ)) becomes a module for the category of line operators, which is given by

L = IndCoh(FlatG(B))

where B = D ∪D× D is the “formal bubble” obtained by gluing two formal disks together along a formal punctureddisk. This monoidal category acts by convolution – double a formal neighbourhood of the point x ∈ Σ and pull-tensor-push along the diagram

FlatG(B)

FlatG(ΣqD× Dx)

qx

OO

p1

vv

p2

((

FlatG(Σ) FlatG(Σ).

In geometric representation theory the category L is called the “spectral Hecke category”. The monoidal unit of Lis given by the skyscraper sheaf at the trivial bundle. If one computes its endomorphism algebra in L one sees ouralgebra A of local operators:

EndL(δ1) ∼= O(h∗[2]/W ) = A.

This isn’t so surprising – morphisms between two line operators should be given by states on a strip compatiblewith these line operators on two sides, and if the line operator on both sides this just gives all states, or all localoperators under the state-operator correspondence.

The upshot to all this is that we obtain our action of the algebra A of local operators as follows.

The action defines a functor L → End(B)

which induces a map A = EndL(δ1)→ EndEndB(idB) = HC•(B)

and therefore a map A→ EndB(F)

for each object F by the universal property of Hochschild cochains. The second line came from the first line byapplying the functor to the algebra of endomorphisms of the unit on each side.

We can now state the main result.

Theorem 3.9 (E-Yoo). The full category of boundary conditions compatible with the vacuum 0 ∈ h∗[2]/W isequivalent to Arinkin and Gaitsgory’s nilpotent singular support category IndCohNG

(FlatG(Σ)).

There’s a simple reason that leads us to expect such a result. There’s a natural map – evaluation at a point x ∈ Σfrom ArthG(Σ) to g∗/G. Post-composing with the eigenvalue map defines a map ArthG(Σ) → h∗/W . The actionof O(h∗[2]/W ) on the category IndCoh(FlatG(Σ)) factors through the natural action of O(ArthG(Σ)) by which wedefine singular support, by pullback along the eigenvalue map (note that this is only a map of ungraded commutativerings, one needs to be more careful to keep track of all the shifts by two). What’s more, the global nilpotent conecan by thought of as coming from the following pullback:

NG //

ArthG(Σ)

evx

0 // h∗/W,

which means being supported at 0 in h∗/W is equivalent to being supported on NG in ArthG(Σ).

17 Section References

3.3 Some Questions and Conjectures

By replacing O(h∗[2]/W ) by its shifted version O(h∗/W ) it makes sense to ask for the category of boundaryconditions compatible with any vacuum v ∈ h∗/W . We can compute this category by a similar method to the oneI just described, and we conjecture that the results fit together in a nice way: that the categories one obtains areequivalent to Arinkin-Gaitsgory categories with the symmetry group broken to a subgroup compatibly with thevacuum. We conjecture the following (and we have some evidence supporting the conjecture).

Conjecture 3.10 (Gauge symmetry breaking). The full subcategory of objects in IndCoh(FlatG(Σ)) compatiblewith the vacuum v ∈ h∗/W is equivalent to IndCohNL

(FlatL(Σ)), where L ⊆ G is the stabilizer of v in g∗.

Remark 3.11. We can prove that the boundary conditions compatible with v ∈ h∗/W are described by a singularsupport condition, namely that the singular support lies in the pullback ArthvG(Σ) = ArthG(Σ)×h∗/W v, definedby the map

ArthG(Σ)evx→ g∗/G→ h∗/W

where the first map evaluates the Arthur parameter at a point x ∈ Σ and the second map is the usual characteristicpolynomial map. The content of the above conjecture is the claim that this singular support condition is equivalentto the NL condition I described. We show that in fact on the level of geometry, not only is ArthvG(Σ) equivalent toNL, but their formal neighbourhoods in ArthG(Σ) and ArthL(Σ) respectively coincide.

Remark 3.12. This conjecture leads us to conjecture something stronger, that there’s a factorization structure onthe Arinkin-Gaitsgory categories. More concretely we make a conjecture about their Hochschild cohomology. Weconjecture that there’s a factorization algebra whose fiber over x ∈ Ran(x) is the direct sum of the algebras⊕

n≥1

⊕x∈Symn(C)

HC•(IndCohNLx(FlatLx

))

where the sum is over x which are lifts of the point x. This factorization structure has a string theoretic origin,coming from the motion of D3 branes in type IIB string theory. There’s an analogous story for D0 branes in acertain twist of type IIA on R2 ×X × C where X is a Calabi-Yau 3-fold: the motion of D0 branes there appearsto produce the factorization structure on cohomological Hall algebras which was constructed by Kontsevich andSoibelman [KS11].

References

[AG12] Dima Arinkin and Dennis Gaitsgory. Singular support of coherent sheaves, and the geometric Langlandsconjecture. arXiv preprint arXiv:1201.6343, 2012.

[BMS07] Rutger Boels, Lionel Mason, and David Skinner. Supersymmetric gauge theories in twistor space.Journal of High Energy Physics, 2007(02):014, 2007.

[Coh76] Fred Cohen. The homology of Cn+1-spaces, n ≥ 0. The homology of iterated loop spaces, pages 207–351,1976.

[Cos13] Kevin Costello. Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4. Pureand Applied Mathematics Quarterly, 9(1):73–165, 2013.

[EY15] Chris Elliott and Philsang Yoo. Geometric Langlands twists of N = 4 gauge theory from derivedalgebraic geometry. arXiv preprint arXiv:1507.03048, 2015.

[EY17] Chris Elliott and Philsang Yoo. A physical origin for singular support conditions in geometric langlandstheory. arXiv prepring arXiv:1707.01292, 2017.

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